Question

Question:

Given: ABCD is a parallelogram and triangle ABD is congruent to triangle DCA.
Prove: ABCD is a rectangle.

Answer 1

It has been proven that ABCD is rectangle, because it is a parallelogram with one right angle.

How to prove that a parallelogram is a rectangle?

A quadrilateral whose diagonals bisect each other is referred to as a parallelogram.

Now, due to the fact that ABCD is a parallelogram, it means that its opposite sides are equal.

Thus:

Triangle ABC ≅ triangle DCB (by SSS congruency postulate)

Thus:

Angle ABC ≅ Angle DCB (CPCTC)

However:

Angle ABC + Angle DCB = 180° (co-interior angles, AB || DC )

Thus:

Angle ABC = Angle DCB = 90°

Hence ABCD is rectangle, because it is a parallelogram with one right angle.